Fist of Fury
Fist of Fury plays in Bad Dudes in each of the game's stages where you're walking across the ground (1, 3, 4, 6, and 7), as opposed to when you're riding (2, and 5) which play Lightning Speed.
The official title come from the Deco History: Data East Game Music Scene One album where the song is named 怒の鉄拳 (Ika no Tekken) which directly translates to "Iron Fist of Anger," but was probably meant to match the Jackie Chan Kung Fu film, "Fist of Fury (怒りの鉄拳)."
Bad Dudes vs. Dragonninja (ARC)
Dragonninja vs. Dragonninja (AMI)
Uses this song for stage 5 instead of Lightning Speed.
Bad Dudes (NES)
The NES port plays the Fist of Fury theme in the same stages as the arcade game.
The song is track 1 in the NSF file.
The Amstrad, C64, and Spectrum ports have different soundtracks that don't include this song. The Apple II, DOS, and MSX ports do not have music.